Why are they called "Bull" or "Bear" markets?

After doing some research it seems that there is no clear consensus as to where these names originally came from but there are two stories that have gained the most traction:

Bearskin traders would sell bearskins they didn’t own (kind of like short selling!), and hope to make a profit by later purchasing skins from trappers at a lower price (the delivery of the skins to the ultimate end use would occur later). So since the bearskin traders made money in a falling market, a falling market became known as a bear market. The term “bull” market comes from the idea that people used to be entertained by bull versus bear fights – thus the bull and bear were mortal enemies and if a bear market meant a falling market, then a bull market must mean the opposite. The practice of bull versus bear fighting was popular both in California and in London during Elizabethan times.

Another explanation posits that a bull attacks by using an upward swinging motion with its horns, and a bear attacks with a downward swipe of its claws on its enemies.